1. Prove that under the transformation of independent variable \( x= \) \( \varphi(t) \), an \( n^{\text {th }} \)-order linear differential equation remains an \( n^{\text {th }} \)-order linear differential equation, and a homogeneous linear differential equation remains a homogeneous linear differential equation. Here \( x=\varphi(t) \) has continuous derivatives up to order \( n \) and \( \varphi^{\prime}(t) \neq 0 \).

1. Find the fourior series of the fanction \( f(x)=\left\{\begin{array}{ll}-1 ; & -1<x<0 \\ 1 ; & 0<x<1\end{array}\right. \)

(6) \( \int \mathrm{e}^{\ln x} \mathrm{~d} x=\ldots \ldots \ldots \ldots+c \) \( \begin{array}{llll}\text { (a) } \frac{x^{2}}{2} \ln x & \text { (b) } \ln x & \text { (c) } \frac{1}{2} x^{2} & \text { (d) } \mathrm{e}^{x}\end{array} \)

2. Evaluate \( \int_{0}^{4 a} \int_{x^{2} / 4 a}^{2 \sqrt{a x}} d y d x \) by changing the order of integration.

41) \( \lim _{x \rightarrow \pi}\left(x^{2}-\pi^{2}\right) \tan \frac{x}{2} \)

53) \( \lim _{x \rightarrow \frac{\pi}{3}} \frac{2 \cos x-1}{\sqrt{3}-\tan x} \)